Problem: Simplify the following expression: $y = \dfrac{9x^2+52x+35}{x + 5}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(9)}{(35)} &=& 315 \\ {a} + {b} &=& &=& {52} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $315$ and add them together. The factors that add up to ${52}$ will be your ${a}$ and ${b}$ When ${a}$ is ${7}$ and ${b}$ is ${45}$ $ \begin{eqnarray} {ab} &=& ({7})({45}) &=& 315 \\ {a} + {b} &=& {7} + {45} &=& 52 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({9}x^2 +{7}x) + ({45}x +{35}) $ Factor out the common factors: $ x(9x + 7) + 5(9x + 7)$ Now factor out $(9x + 7)$ $ (9x + 7)(x + 5)$ The original expression can therefore be written: $ \dfrac{(9x + 7)(x + 5)}{x + 5}$ We are dividing by $x + 5$ , so $x + 5 \neq 0$ Therefore, $x \neq -5$ This leaves us with $9x + 7; x \neq -5$.